Solve frac{sqrt{3}}{2+sqrt{3}}

$$\frac{ \sqrt{ 3 } }{ 2+ \sqrt{ 3 } }$$

Evaluate

$$2\sqrt{3}-3\approx 0.464101615$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{2+\sqrt{3}} by multiplying numerator and denominator by 2-\sqrt{3}.

$$\frac{\sqrt{3}\left(2-\sqrt{3}\right)}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}$$

Consider \left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{3}\left(2-\sqrt{3}\right)}{2^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 2. Square \sqrt{3}.

$$\frac{\sqrt{3}\left(2-\sqrt{3}\right)}{4-3}$$

Subtract 3 from 4 to get 1.

$$\frac{\sqrt{3}\left(2-\sqrt{3}\right)}{1}$$

Anything divided by one gives itself.

$$\sqrt{3}\left(2-\sqrt{3}\right)$$

Use the distributive property to multiply \sqrt{3} by 2-\sqrt{3}.

$$2\sqrt{3}-\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$2\sqrt{3}-3$$